2. It is required to find the fifth power of the re

a3 3 a2 b + 3 a 62-63, third power.

b2 — b3,

b2

a44a3 b+6 a2 b2 - 4 a b3 + 64, fourth power

a4a4 b6 a3 b2 4 a2 b3 + a b1

a4b+4 a3 b2-6 a2 b3+4 a b4-b5

5 a1 b+10 a3 b2 —10 a2 b3 + 5 a ba — b3, Ans

3. What is the second power of 3 ab+2 c?

+6ac2b c + 4 c2

9a2-6 ab 12 a c + b2 - 4 b c + 4 c2.

power.

4. What is the third power of a + 1? 5. Raise x + y + z to the second power. 6. Involve a+b+c+d to the second 7. What is the third power of 2 a x + c2 ? 8. Involve 3 a + 2 b to the third power. 9. What is the second power of a + b — c? 10. What is the third power of a

+x

11. What is the second power of ?

For=

a a

b b

a2

8?

ANS..

or, according to the princi

b2

ples already explained. Therefore, we involve a fraction, by raising both the numerator and denominator to the required power.

15. What is the second power of a+b?

X

16. What is the third power of a=5?

a3

20. Required the second power of 5-43

21. What is the second power of "?

a2b

SECTION III.

The Binomial Theorem.

The first and second examples of the last section exhibit the method of involving binomial and residual quantities, by actual multiplication. It is very apparent that, when a high power is required, the process must become tedious. From these operations, however, may be derived rules for raising such quantities to any proposed power, without the intervention of the lower powers. This method of involving quantities was discovered by Sir Isaac Newton, and is called the Binomial Theorem.

When we would express any required power of a given binomial or residual quantity, four things require attention; namely, the number of terms in that power, the signs, the exponents, and the coefficients.

THE TERMS. It will be observed, that the number of terms in every power, is greater by one than the exponent of that power. Thus, the second power consists of three terms; the third power, of four terms; the fourth power, of five terms, &c.

THE SIGNS. The corresponding powers of these two quantities, a + b and a — b, differ only in their signs. All the terms of the binomial quantity are positive; whereas the EVEN terms of the residual quantity are negative, and the ODD terms positive.

THE EXPONENTS. The first term of every power consists of the first letter of the given binomial, a,

raised to that power; and the other exponents of that letter diminish by one towards the right. The first letter of the binomial, a, is called the leading quantity; and its exponents for the seventh power, are

7, 6, 5, 4, 3, 2, 1.

The last term of every power is the last letter of the given binomial, b, raised to that power; and the other exponents of that letter diminish by unity towards the left. The last letter of the binomial, b, is called the following quantity; and its exponents for the seventh

power, are

1, 2, 3, 4, 5, 6, 7.

The sum of the exponents of the two letters, is always equal to the exponent of the power in which they occur.

To apply these principles, let it be required to determine the number of terms, the signs and exponents of a + b and a b, in the seventh power. The number of terms will be eight, that is, one more than the power proposed; and their signs and exponents, without the coefficients, will stand thus: (a+b)"=a2+ab+a5 b2+a1 b3+a3b1+a2b5+ab®+b7 (a—b)7a7—a® b+a5 b2—a1b3+a3 b1—a2 b5 + a b¤—b7

THE COEFFICIENTS. It will be seen, by an inspec tion of the examples above referred to, that the coeffi cient of the first term is always 1, and that the coefficient of the second term is the exponent of the power. The remaining coefficients may be found by the following RULE: If the coefficient of any term be multiplied by the exponent of the leading quantity in that

term, and the product be divided by the number which denotes its place from the left, it will give the coeffi cient of the next term.

The coefficients of a + b, raised to the seventh power, are

If we prefix these coefficients to the terms of a + b b, already obtained, we shall have the seventh powers of those quantities complete :

and a

(a + b)2 = a2 + 7 a® b + 21 a5 b2 + 35 a4 b3 + 35 a3 b4 + 21 a2 b5 + 7 a bε + 67.

(a - b)7 = a7 — 7 a b + 21 a5 b2 — 35 a4 b3 + 35 a3 b4 — 21 a2 b5 +7 a b¤ — b7.

It will be observed, that the coefficients are equal in the first and last terms, also in the second and last but one, the third and last but two, and so on. It will be sufficient, therefore, in practice, to find the coefficients of half the terms, if their number be even, or of one more than half, if it be odd, and apply them to the rest.

3. What is the fourth power of x+y?

ANS. x+4x3 y + 6 x2 y2 + 4 x y3 + y1

4. Raise x y to the sixth power.

5. What is the fifth power of a + c? 6. What is the eighth power of mn? 7. What is the fourth power of m+n? 8. Required the seventh power of x + y. 9. What is the ninth power of a + 6? 10. Raise cd to the tenth power.